Proper use of "while" vs "as" in a sentence?

This can get complicated, but the basic idea is that while=during, and as=coincident with.  During implies that the things are somehow linked and the end of one requires the end of the other.  Coincidence implies that there is no link of importance.  Neither one really has anything to do with the other except that both happen at the same time.

I would say "while his eyes glow", because presumably if his eyes are not glowing, he is not communicating telepathically.  The glowing is required for the communication.

In the second example, I think you could use either.  While would mean that the falling and the dropping are linked events, and the one is the cause of the other (presumably falling made her drop the weapon). As would mean that there really was no link between the two, that both were caused by a third event (in this case) and neither had to happen simply because the other happened.  SHe dropped her weapon, and at the same time also tumbled.  She did not drop the weapon because she tumbled, and she did not tumble because she dropped the weapon.  They just both happened to occur at the same moment.

We can use while or as to talk about two longer events or activities happening at the same time. We can use either simple or continuous verb forms:

We spent long evenings talking in my sitting-room while he played the music he had chosen and explained his ideas.

We were lying on the beach sunbathing as they were playing volleyball.